A memory buffer selecting between a parallel test mode and a mode register control mode, and a memory module and memory system having the memory buffer are disclosed. The memory buffer includes a control circuit and a mode selecting circuit. The control circuit generates a mode control signal based on a first chip selecting signal, a second chip selecting signal, a row address signal, a column address signal, and a write enable signal. The mode selecting circuit selects one of a parallel test mode and a mode register control mode in response to the mode control signal.